The television EHT is an ordinary flyback transformer, one of the two common types of transformer we meet in engineering.
The 12v main power DC is not simply put across its input, but is chopped by semiconductor switches.
The EHT transformer will have a much higher ratio of turns than say the flyback in a switch mode power supply.
The output diode is special in some sense, in that it is built to withstand working with many 10s of kV. The limit for a practical single silicon junction diode is somewhere between 1kV and 2kV. The EHT diodes used in a TV are a series stack of many diode die placed in series, and then sealed in the same package. If you do manage to get your hands on an EHT diode (they tend to be potted inside the EHT transformer for insulation and cost), then when you measure the forward voltage drop you will find it's often well above 10v, the result of it being many individual diodes inside.